Pogba was taken off in the 19th minute of the game at Old Trafford after over-stretching.
The Manchester United midfielder initially tried to carry on but then signalled to the bench that he could not carry on and was replaced by Marouane Fellaini.
Former United star Owen Hargreaves, who was commentating on the game, says it is a huge blow for United boss Jose Mourinho.
Hargreaves says Pogba is the only player in the squad who cannot be replaced.
"He’s something special, hopefully he can just run this one off,” he said on BT Sport.
Pogba could now miss United's visit of Everton this Sunday in the Premier League (4pm).
He would also be a doubt for the EFL Cup tie with Burton Albion, the trip to Southampton and the next Champions League tie away to CSKA Moscow.
The man who replaced Paul Pogba, Fellaini, opened the scoring for the hosts against Basel tonight 16 minutes after coming on.
United went on to win the game 3-0, with Romelu Lukaku and Marcus Rashford sealing a relatively comfortable win.
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